On the Measurement of Qubits

Home , Stokes parameters

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The ability to create, manipulate and characterize quantum states is becoming an increasingly important area of physical research, with implications for areas of technology such as quantum computing, quantum cryptography and communications. With a series of measurements on a large enough number of identically prepared copies of an quantum system, one can infer,to a reasonable approximation, the quantum state of the system. Arguably, the first such experimental technique for determining the state of quantum system was devised by George Stokes in 1852 [1]. His famous four parameters allow an experimenter to determine uniquely the polarization state of a light beam. With the insight provided by nearly 150 years of progress in optical physics, we can consider coherent light beams to be an ensemble of two-level quantum mechanical systems, the two levels being the two polarization degrees of freedom of the photons; the Stokes parameters allow one to determine the density matrix describing this ensemble. More recently, experimental techniques for the measurement of the more subtle quantum properties of light have been the subject of intensive investigation (see ref. [2] for a comprehensive and erudite exposition of this subject). In various experimental circumstances is has been found reasonably straightforward to devise a simple linear tomographic technique in which the density matrix (or Wigner function) of a quantum state is found from a linear transformation of experimental data. However, there is one important drawback to this method, in that the recovered state might not correspond to a physical state because of experimental noise. For example, density matrices for any quantum state must be Hermitian, positive semi-definite matrices with unit trace. The tomographically measured matrices often fail to be positive semi-definite, especially when measuring low-entropy states. To avoid this problem the “maximum likelihood” tomographic approach to the estimation of quantum states has been developed [3–7]. In this approach the density matrix that is “mostly likely” to have produced a measured data set is determined by numerical optimization. In the past decade several groups have successfully employed tomographic techniques for the measurement of quantum mechanical systems. In 1990 Risley et al. at North Carolina State University reported the measurement of the density matrix for the nine sublevels of the n = 3 level of hydrogen atoms formed following collision between H+ ions and He atoms, in conditions of high symmetry which simplified the tomographic problem [8]. Since then, in 1993 Smithey et al. (University of Oregon), made a homodyne measurement of the Wigner function of a single mode of light [9]. Other explorations of the quantum states of single mode light fields have been made by Mlynek et al. (University of Konstanz, Germany) [10] and Bachor et al. (Australian National University) [11]. Other quantum systems whose density matrices have been investigated experimentally include the vibrations of molecules [12], the motion ions and atoms [13,14] and the internal angular momentum quantum state of the F = 4 ground state of a cesium atom [15]. The quantum states of multiple spin-half nuclei have been measured in the high-temperature regime using NMR techniques [16], albeit in systems of such high entropy that the creation of entangled states is necessarily precluded [17]. The measurement of the quantum state of entangled qubit pairs, realized using the polarization degrees of freedom of a pair of photons created in a parametric down-conversion experiment was reported by us recently [18]. In this paper we will examine techniques in detail for quantum state measurement as it applies to multiple correlated two-level quantum mechanical systems (or “qubits” in the terminology of quantum information). Our particular emphasis is qubits realized via the two polarization degrees of freedom of photons, data from which we use to illustrate our results. However, these techniques are readily applicable to other technologies proposed for creating entangled states of pairs of two-level systems. Because of the central importance of qubit systems to the emergent discipline of quantum computation, a thorough explanation of the techniques needed to characterize the qubit states will be of relevance to workers in the various diverse experimental fields currently under consideration for quantum computation technology [19]. This paper is organized as follows: In Section II we explore the analogy with the Stokes parameters, and how they lead naturally to a scheme for measurement of an arbitrary number of two-level systems. In Section III, we discuss the measurement of a pair of qubits in more detail, presenting the validity condition for an arbitrary measurement scheme and introducing the set of 16 measurements employed in our experiments. Section IV deals with our method for maximum likelihood reconstruction and in Section V we demonstrate how to calculate the errors in such measurements, and how these errors propagate to quantities calculated from the density matrix.

II. THE STOKES PARAMETERS AND QUANTUM STATE TOMOGRAPHY

As mentioned above, there is a direct analogy between the measurement of the polarization state of a light beam and the measurement of the density matrix of an ensemble of two-level quantum mechanical systems. Here we explore this analogy in more detail.

2 A. Single qubit tomography

The Stokes parameters are deﬁned from a set of four intensity measurements [20] : (i) with a ﬁlter that transmits 50% of the incident radiation, regardless of its polarization; (ii) with a polarizer that transmits only horizontally polarized light; (iii) with a polarizer that transmits only light polarized at 45o to the horizontal; and (iv) with a polarizer that transmits only right circularly polarized light. The number of photons counted by a detector, which is proportional to the classical intensity, in these four experiments are as follows:

n = N ( H ρˆ H + V ρˆ V )= N ( R ρˆ R + L ρˆ L ) 0 2 h | | i h | | i 2 h | | i h | | i n = ( H ρˆ H ) 1 N h | | i = N ( R ρˆ R + R ρˆ L + L ρˆ R + L ρˆ L ) 2 h | | i h | | i h | | i h | | i n = D¯ ρˆ D¯ 2 N h | | i = N ( R ρˆ R + L ρˆ L i L ρˆ R + i R ρˆ L ) 2 h | | i h | | i− h | | i h | | i n = ( R ρˆ R ) 3 N h | | i (2.1)

Here H , V , D¯ = ( H V ) /√2 = exp(iπ/4) ( R + i L ) /√2 and R = ( H i V ) /√2 are the kets representing| i qubits| i | polarizedi | i in − | thei linear horizontal, linea| i r vertical,| i linear| diagonali | i− (45o)| andi right-circular senses respectively,ρ ˆ is the (2 2) density matrix for the polarization degrees of the light (or, for a two-level quantum system) and is constant× dependent on the detector eﬃciency and light intensity. The Stokes parameters, which fully characterizeN the polarization state of the light, are then deﬁned by

2n = ( R ρˆ R + L ρˆ L ) S0 ≡ 0 N h | | i h | | i 2 (n n )= ( R ρˆ L + L ρˆ R ) S1 ≡ 1 − 0 N h | | i h | | i 2 (n n )= i ( R ρˆ L L ρˆ R ) S2 ≡ 2 − 0 N h | | i − h | | i 2 (n n )= ( R ρˆ R L ρˆ L ) . S3 ≡ 3 − 0 N h | | i − h | | i (2.2)

We can now relate the Stokes parameters to the density matrix ρˆ by the formula

3 1 i ρˆ = S σˆ , (2.3) 2 i i=0 0 X S whereσ ˆ = R R + L L is the single qubit identity operator andσ ˆ = R L + L R ,σ ˆ = i( L R R L and 0 | ih | | ih | 1 | ih | | ih | 2 | ih | − | ih | σˆ3 = R R L L are the Pauli spin operators. Thus the measurement of the Stokes parameters can be considered equivalent| ih to| − a | tomographicih | measurement of the density matrix of an ensemble of single qubits.

B. Multiple beam Stokes parameters: multiple qubit tomography

The generalization of the Stokes scheme to measure the state of multiple photon beams (or multiple qubits) is reasonably straightforward. One should, however, be aware that importance diﬀerences exist between the one-photon and the multiple photon cases. Single photons, at least in the current context, can be described in a purely classical manner, and the density matrix can be related to the purely classical concept of the coherency matrix [21]. For multiple photon one has the possibility of non-classical correlations occurring, with quintessentially quantum-mechanical phenomena such as entanglement being present. We will return to the concept of entanglement and how it may be measured later in this paper. An n-qubit state is characterized by a density matrix which may be written as follows:

3 1 ρˆ = ri ,i ,...i σî σî . . . σî , (2.4) 2n 1 2 n 1 ⊗ 2 ⊗ ⊗ n i ,i ,...i =0 1 2Xn

3 n where the 4 parameters ri1,i2,...in are real numbers. The normalization property of the density matrices requires that n r0,0,...0 = 1, and so the density matrix is specified by 4 1 real parameters. The symbol represents the tensor product between operators acting on the Hilbert spaces asso− ciated with the separate qubits. ⊗ As Stokes showed, the state of a single qubit can be determined by taking a set of four projection measurements which are represented by the four operatorsµ ˆ0 = H H + V V ,µ ˆ1 = H H ,µ ˆ2 = D¯ D¯ ,µ ˆ3 = R R . Similarly, the state of two qubits can be determined| byih the| se|tih of 16| measurements| ih | represented| ih by| the operators| ih | n µî µˆj (i, j = 0, 1, 2, 3). More generally the state of an n-qubit system can be determined by 4 measurements ⊗ given by the operatorsµ î1 µî2 . . . µîn (ik =0, 1, 2, 3) and (k =1, 2,...n). This ‘tree’ structure for multi-qubit measurement is illustrated⊗ in fig.1.⊗ ⊗ The proof of this conjecture is reasonably straightforward. The outcome of a measurement is given by the formula

n = Tr ρˆµˆ , (2.5) N { } whereρ ˆ is the density matrix,µ ˆ is the measurement operator and is a constant of proportionality which can be determined from the data. Thus in our n-qubit case, the outcomes ofN the various measurement are

ni ,i ,...i = Tr ρˆ(ˆµi µˆi . . . µˆi ) . (2.6) 1 2 n N { 1 ⊗ 2 ⊗ ⊗ n } Substituting from eq.(2.4) we obtain

3 ni ,i ,...i = N Tr µî σˆj Tr µî σˆj . . . Tr µî σˆj ri ,i ,...i . (2.7) 1 2 n 2n { 1 1 } { 2 2 } { n n } 1 2 n j ,j ,...j =0 1 2Xn

As can be easily verified, the single qubit measurement operatorsµ î are linear combinations of the Pauli operators 3 σˆj , i.e.µ î = j=0 Υij σˆj , where Υij are the elements of the matrix

P 1 0 0 0 1/2 1/20 0 Υ= . (2.8)  1/2 0 1/2 0   1/20 01/2      Further, we have the relation Tr σîσˆj =2δij (where δij is the Kronecker delta). Hence eq.(2.7) becomes { } 3 ni ,i ,...i = Υi j Υi j . . . Υi j ri ,i ,...i . (2.9) 1 2 n N 1 1 2 2 n n 1 2 n j ,j ,...j =0 1 2Xn 3 1 Introducing the left-inverse of the matrix Υ, defined so that k=0(Υ− )ikΥkj = δij and whose elements are 1 000P 1200 Υ 1 = , (2.10) −  −1020  −  1002   −    we can find a formula for the parameters ri1 ,i2,...in in terms of the measured quantities ni1,i2,...in , viz.:

3 1 1 1 ri1 ,i2,...in = Υ− Υ− . . . Υ− ni1,i2,...in N i1j1 i2j2 injn j ,j ,...jn=0 1 2X i ,i ,...i . (2.11) ≡ S 1 2 n

In eq.(2.11) we have introduced the n-photon Stokes parameter i1,i2,...in , deﬁned in an analogous manner to the single photon Stokes parameters give in eq.(2.2). S Since, as already noted, r0,0,...0 = 1, we can make the identiﬁcation 0,0,...0 = , and so the density matrix for the n-qubit system can be written in terms of the Stokes parameters as follows:S N

3 1 i1,i2,...in ρˆ = n S σî1 σî2 . . . σîn . (2.12) 2 , ,... ⊗ ⊗ ⊗ i ,i ,...i =0 0 0 0 1 2Xn S

4 This is a recipe for measurement of the density matrices which, assuming perfect experimental conditions and the complete absence of noise, will always work. It is important to realize that the set of four Stokes measurements µˆ , µˆ , µˆ , µˆ are not unique: there may be circumstances in which it is more convenient to use some other set, { 0 1 2 3} which are equivalent. A more typical set, at least in optical experiments, isµ ˆ′ = H H ,µ ˆ′ = V V ,µ ˆ′ = D D , 0 | ih | 1 | ih | 2 | ih | µˆ3′ = R R . In the| ih following| section we will explore more general schemes for the measurement of two qubits, starting with a discussion, in some detail, of how the measurements are actually performed.

III. GENERALIZED TOMOGRAPHIC RECONSTRUCTION OF THE POLARIZATION STATE OF TWO PHOTONS

A. Experimental set-up

The experimental arrangement used in our experiments is shown schematically in Fig.1. An optical system consisting of lasers, polarization elements and non-linear optical crystals (and collectively characterized for the purposes of this paper as a “black-box”,) is used to generate pairs of qubits in an almost arbitrary quantum state of their polarization degrees of freedom. A full description of this optical system and how such quantum states can be prepared can be found in ref. [22–24] † The output of the black box consists of a pair of beams of light, whose quanta can be measured by means of photo-detectors. To project the light beams onto a polarization state of the experimenter’s choosing, three optical elements are placed in the beam in front of each detector: a polarizer (which transmits only vertically polarized light), a quarter-wave plate and a half-wave plate. The angles of the fast axes of both of the waveplates can be set arbitrarily, allowing the V projection state ﬁxed by the polarizer to be rotated into any polarization state that the experimenter may wish.| i Using the Jones calculus notation, with the following convention,

0 1 = V , = H , (3.1) 1 | i 0 | i where V ( H ) is the ket for a vertically (horizontally) polarized beam, the eﬀect of quarter and half wave plates whose fast| i axes| i are at angles q and h with respect to the vertical axis, respectively, are given by the 2 2 matrices × 1 i cos(2q) sin(2q) UˆQW P (q)= − , √2 sin(2q) i + cos(2q) cos(2h) sin(2h) Uˆ (q)= . (3.2) HW P sin(2h) −cos(2h) − − Thus the projection state for the measurement in one of the beams is given by

(1) 0 ψ (h, q) = UˆQW P (q) UˆHW P (h) | proj i · · 1 = a(h, q) H + b(h, q) V , (3.3) | i | i where, neglecting an overall phase, the functions a(h, q) and b(h, q) are given by 1 a(h, q)= (sin(2h) i sin[2(h q)]) , √2 − − 1 b(h, q)= (cos(2h)+ i cos[2(h q)]) . (3.4) −√2 − The projection state for the two beams is given by

†It is important to realize that the entangled photon pairs are produced in a non-deterministic manner: one cannot specify with certainly when a photon pair will be emitted; indeed there is a small probability of generating four, or six or higher numbers of photons. Thus we can only post-selectively generate entangled photon pairs: i.e. one only knows that the state was created after if has been measured.

We shall denote the projection state corresponding to one particular set of waveplate angles h ,ν , q ,ν ,h ,ν , q ,ν { 1 1 2 2 } ‡ by the ket ψν ; thus the projection measurement is represented by the operatorµ ˆν = ψν ψν . Consequently, the average number| i of coincidence counts that will be observed in a given experimental run is| ih |

nν = ψν ρˆ ψν (3.6) N h | | i whereρ ˆ is the density matrix describing the ensemble of qubits, and is a constant dependent on the photon flux N and detector efficiencies. In what follows, it will be convenient to consider the quantities sν defined by

sν = ψν ρˆ ψν . (3.7) h | | i

B. Tomographically Complete set of Measurements

In Section II we have given one possible set of projection measurements ψν ψν which which uniquely determine the density matrixρ ˆ. However, one can conceive of situations in which these{| i will h |}not be the most convenient set of measurements to make. Here we address the problem of ﬁnding other sets of suitable measurements. The smallest number of states required for such measurements can be found by a simple argument: there are 15 real unknown parameters which determine a 4 4 density matrix, plus there is the single unknown real parameter , making a total of 16. × N In order to proceed it is helpful to convert the 4 4 matrixρ ˆ into a 16-dimensional column vector. To do this we × use a set of 16 linearly independent 4 4 matrices Γˆν which have the following mathematical properties: × { }

Tr Γˆν Γˆµ = δν,µ · n o 16 Aˆ = Γˆν Tr Γˆν Aˆ A,ˆ (3.8) · ∀ ν=1 X n o where Aˆ is an arbitrary 4 4 matrix. Finding a set of Γˆν matrices is in fact reasonably straightforward: for example, the set of (appropriately× normalized) generators of the Lie algebra SU(2) SU(2) fulﬁll the required criteria (for reference, we list this set in Appendix A). These matrices are of course simply⊗ a re-labeling of the two-qubit Pauli matricesσ ˆi σˆj (i, j =0, 1, 2, 3) discussed above. Using these matrices the density matrix can be written as ⊗ 16 ρˆ = Γˆν rν , (3.9) ν=1 X where rν is the ν-th element of a sixteen element column vector, given by the formula

rν = Tr Γˆν ρˆ (3.10) · n o Substituting from eq.(3.9) into eq.(3.6), we obtain the following linear relationship between the measured coincidence counts nν and the elements of the vector rµ:

16 nν = Bν,µrµ (3.11) N µ=1 X

‡Here the ﬁrst subscript on the waveplate angle refers one of the two photon beams; the second subscript distinguishes which of the sixteen diﬀerent experimental states is under consideration.

6 where the 16 16 matrix Bν,µ is given by ×

Bν,µ = ψν Γˆµ ψν . (3.12) h | | i

Immediately we ﬁnd a necessary and suﬃcient condition for the completeness of the set of tomographic states ψν : {| i} if the matrix Bν,µ is non- singular, then eq.(3.11) can be inverted to give

16 1 1 rν = ( )− B− nµ. (3.13) N ν,µ µ=1 X The set of sixteen tomographic states which we employed are given in Table 1. They can be shown to satisfy the condition that Bν,µ is non- singular. By no means are these states unique in this regard: these were the states chosen principally for experimental convenience. These states can be realized by setting specific values of the half- and quarter-wave plate angles. The appropriate values of these angles (measured from the vertical) are given in Table 1. Note that overall phase factors do not affect the results of projection measurements. Substituting eq.(3.13) into eq.(3.9), we find that

16 1 ρˆ = ( )− Mˆ ν nν N ν=1 X 16 = Mˆ ν sν , (3.14) ν=1 X where the sixteen 4 4 matrices Mˆ ν are deﬁned by × 16 ˆ 1 ˆ Mν = B− ν,µ Γµ. (3.15) ν=1 X The introduction of the Mˆ ν matrices allows a compact form of linear tomographic reconstruction, eq.(3.14), will be most useful in the error analysis that follows. These Mˆ ν matrices, valid for our set of tomographic states, are listed in Appendix B, together with some of their important properties. We can use one of these properties, eq.(B.6), to obtain the value of the unknown quantity . That relationship implies N

Tr Mˆ ν ψν ψν ρˆ =ρ. ˆ (3.16) ν | ih | X n o Taking the trace of this formula, and multiplying by we obtain: N

Tr Mˆ ν nν = . (3.17) ν N X n o For our set of tomographic states, it can be shown that

1 if ν =1, 2, 3, 4 Tr Mˆ = (3.18) ν 0 if ν =5,... 16, ν X n o hence the value of the unknown parameter in our experiments is given by: N 4 = nν N ν=1 X = ( HH ρˆ HH + HV ρˆ HV + V H ρˆ V H + V V ρˆ V V ) . (3.19) N h | | i h | | i h | | i h | | i Thus we obtain the ﬁnal formula for the tomographic reconstruction of the density matrices of our states:

16 4 ρˆ = Mˆ ν nν / nν . (3.20) ν=1 ! ν=1 ! X X

7 As an example, the following set of 16 counts were taken for the purpose of tomographically determining the density matrix for an ensemble of qubits all prepared in a speciﬁc quantum state: n1 = 34749,n2 = 324,n3 = 35805,n4 = 444,n5 = 16324,n6 = 17521,n7 = 13441,n8 = 16901,n9 = 17932,n10 = 32028,n11 = 15132,n12 = 17238,n13 = 13171,n14 = 17170,n15 = 16722,n16 = 33586 Applying eq.(3.20) we ﬁnd

0.4872 0.0042 + i0.0114 0.0098 i0.0178 0.5192 + i0.0380 0.0042 i0.0114− 0.0045− 0.0271 −i0.0146 0.0648 i0.0076 ρˆ = (3.21)  −0.0098 +− i0.0178 0.0271 + i0.0146 0.0062− −0.0695 +− i0.0134  − −  0.5192 i0.0380 0.0648 + i0.0076 0.0695 i0.0134 0.5020   − − − −    This matrix is shown graphically in fig.3a. Note that, by construction, the density matrix is normalized, i.e. Tr ρˆ = 1 and Hermitian, i.e.ρ ˆ† =ρ ˆ . However, when one calculates the eigenvalues of this measured density matrix,{ } one finds the values 1.02155, 0.0681238, - 0.065274 and -0.024396; and also, Tr ρˆ2 = 1.053 . Density matrices for all physical states must have the property of positive semi-definiteness, which (in{ conjunction} with the normalization and Hermiticity properties) imply that all of the eigenvalues must lie in the interval [0, 1], their sum being 1; this in turn implies that 0 Tr ρˆ2 1. Clearly the density matrix reconstructed above by linear tomography violates these condition. From≤ our experience{ } ≤ of tomographic measurements of various mixed and entangled states prepared experimentally, this seems to happen roughly 75% of the time for low entropy, highly entangled states; it seems to have a higher probability of producing the correct result for states of higher entropy, but the cautious experimenter should check every time. The obvious culprit for this problem is experimental inaccuracies and statistical fluctuations of coincidence counts, which mean that the actual numbers of counts recorded in a real experiment differ from those that can be calculated by eq.(3.6). Thus the linear reconstruction is of limited value for states of low entropy (which are of most experimental interest because of their application to quantum information technology); however as we shall see, the linear approach does provide a useful starting point for the numerical optimization approach to density matrix estimation which we will discuss in the next section.

IV. MAXIMUM LIKELIHOOD ESTIMATION

As mentioned in Section III, the tomographic measurement of density matrices can produce results which violate important basic properties such as positivity. To avoid this problem, the maximum likelihood estimation of density matrices may be employed. Here we describe a simple realization of this technique.

A. Basic approach

Our approach to the maximum likelihood estimation of the density matrix is as follows: (i) Generate a formula for an explicitly “physical” density matrix, i.e. a matrix which has the three important properties of normalization, Hermiticity and positivity. This matrix will be a function of 16 real variables (denoted t ,t ,...t ). We will denote the matrix asρ ˆp(t ,t ,...t ). { 1 2 16} 1 2 16 (ii) Introduce a “likelihood function” which quantiﬁes how good the density matrixρ ˆp(t1,t2,...t16) is in relation to the experimental data. This likelihood function is a function of the 16 real parameters tν and of the 16 experimental data nν . We will denote this function as (t ,t ,...t ; n ,n ,...n ). L 1 2 16 1 2 16 (iii) Using standard numerical optimization techniques, ﬁnd the optimum set of variables t(opt),t(opt),...t(opt) for { 1 2 16 } which the function (t1,t2,...t16; n1,n2,...n16) has its maximum value. The best estimate for the density matrix is (opt) (opt) L (opt) thenρ ˆ(t1 ,t2 ,...t16 ). The details of how these three steps can be carried out are described in the next three sub-sections.

B. Physical Density Matrices

The property of non-negative deﬁniteness for any matrix ˆ is written mathematically as G ψ ˆ ψ 0 ψ . (4.1) h |G| i≥ ∀| i

8 Any matrix that can be written in the form ˆ = Tˆ†Tˆ must be non-negative deﬁnite. To see that this is the case, substitute into eq.(4.1): G

ψ Tˆ†Tˆ ψ = ψ′ ψ′ 0, (4.2) h | | i h | i≥ where we have deﬁned ψ′ = Tˆ ψ . Furthermore (Tˆ†Tˆ)† = Tˆ†(Tˆ†)† = Tˆ†Tˆ, i.e. ˆ = Tˆ†Tˆ must be Hermitian. To ensure normalization, one| i can simply| i divide by the trace: thus the matrixg ˆ given byG the formula

gˆ = Tˆ†T/ˆ Tr Tˆ†Tˆ (4.3) { } has all three of the mathematical properties which we require for density matrices. For the two qubit system, we have a 4 4 density matrix with 15 independent real parameters. Since it will be useful to be able to invert relation (4.3), it× is convenient to choose a tri-diagonal form for Tˆ:

t1 0 00 t + it t 0 0 Tˆ(t)=  5 6 2  , (4.4) t11 + it12 t7 + it8 t3 0  t15 + it16 t13 + it14 t9 + it10 t4      Thus the explicitly “physical” density matrixρ ˆp is given by the formula

ρˆp(t)= Tˆ†(t)Tˆ(t)/Tr Tˆ†(t)Tˆ(t) . (4.5) { } For future reference, the inverse relationship, by which the elements of Tˆ can be expressed in terms of the elements ofρ ˆ, is as follows:

∆ (1) 0 00 M11  q  (1) (1) M12 M11  (1) (2) (2) 0 0   11,22   M11 M11,22 r M  Tˆ =   . (4.6)  p   (2) (2) (2)   M12,23 M11,23 M11,22 0   (2) (2) ρ44   √ρ44 11,22 √ρ44 11,22   M M r     pρ41 pρ42 ρ43   √ρ44   √ρ44 √ρ44 √ρ44    (1) Here we have used the notation ∆ = Det(ˆρ); ij is the ﬁrst minor ofρ ˆ, i.e. the determinant of the 3 3 matrix M (2) × formed by deleting the i-th row and j-th column ofρ ˆ; ij,kl is the second minor ofρ ˆ, i.e. the determinant of the 2 2 matrix formed by deleting the i-th and k-th rows andM j-th and l-th columns ofρ ˆ (i = k and j = l). × 6 6

C. The Likelihood Function

The measurement data consists of a set of 16 coincidence counts nν (ν = 1, 2,... 16) whose expected value is n¯ν = ψν ρˆ ψν . Let us assume that the noise on these coincidence measurements has a Gaussian probability distribution.N h | Thus| i the probability of obtaining a set of 16 counts n ,n ,...N is { 1 2 16} 16 2 1 (nν n¯ν ) P (n ,n ,...n )= exp − , (4.7) 1 2 16 Norm − 2σ2 ν=1 ν Y where σν is the standard deviation for ν-th coincidence measurement (given approximately by √n¯ν ) and Norm is the normalization constant. For our physical density matrixρ ˆp the number of counts expected for the ν-th measurement is

n¯ν (t ,t ,...t )= ψν ρˆp (t ,t ,...t ) ψν . (4.8) 1 2 16 N h | 1 2 16 | i

Thus the likelihood that the matrixρ ˆp (t ,t ,...t ) could produce the measured data n ,n ,...n is 1 2 16 { 1 2 16}

9 16 2 1 ( ψν ρˆp (t1,t2,...t16) ψν nν ) P (t1,t2,...t16)= exp N h | | i− , (4.9) Norm − 2 ψν ρˆp (t ,t ,...t ) ψν nν ) ν=1 1 2 16 Y N h | | i− 4 where = Nν . N ν=1 Rather than find maximum value of P (t1,t2,...t16) it simplifies things somewhat to find the maximum of its P logarithm (which is mathematically equivalent) §. Thus the optimization problem reduces to finding the minimum of the following function:

16 2 ( ψν ρˆp (t1,t2,...t16) ψν nν ) (t1,t2,...t16)= N h | | i− . (4.10) L 2 ψν ρˆp (t ,t ,...t ) ψν nν ) ν=1 1 2 16 X N h | | i− This is the “likelihood” function which we employed in our numerical optimization routine.

D. Numerical Optimization

We used the Mathematica 4.0 routine FindMinimum which executes a multidimensional Powell direction set algorithm (see ref. [25] for a description of this algorithm). To execute this routine, one requires an initial estimate for the values of t1,t2,...t16. For this, we used the tomographic estimate of the density matrix in the inverse relation (4.6), allowing us to determine a set of values for t1,t2,...t16. Since the tomographic density matrix may not be non-negative deﬁnite, the values of the tν ’s deduced in this manner are not necessarily real. Thus for our initial guess we used the real parts of the tν ’s deduced from the tomographic density matrix. For the example given in Section 2, the maximum likelihood estimate is

0.5069 0.0239 + i0.0106 0.0412 i0.0221 0.4833 + i0.0329 0.0239 i0.0106− 0.0048− 0.0023 +−i0.0019 0.0296 i0.0077 ρˆ = . (4.11)  −0.0412 +− i0.0221 0.0023 i0.0019 0.0045 −0.0425− + i0.0192  − − −  0.4833 i0.0329 0.0296 + i0.0077 0.0425 i0.0192 0.4839   − − − −    This matrix is illustrated in Fig.2b. In this case, the matrix has eigenvalues 0.986022, 0.0139777, 0 and 0; and Tr ρˆ2 = 0.972435, indicating, while the linear reconstruction gave a non-physical density matrix, the maximum- likelihood{ } reconstruction gives a legitimate density matrix.

V. ERROR ANALYSIS

In this section we present an analysis of the errors inherent in the tomographic scheme described in Section III. Two sources of errors are found to be important: the shot noise error in the measured coincidence counts nν and the uncertainty in the settings of the angles of the waveplates used to make the tomographic projection states. We will analyze these two sources separately. In addition to determining the density matrix of a pair of qubits, one is often also interested in quantities derived from the density matrix, such as the entropy or the entanglement of formation. For completeness, we will also derive the errors in some of these quantities.

A. Errors due to Count Statistics

From eq.(3.20) we see that the density matrix is speciﬁed by a set of sixteen parameters sν deﬁned by

sν = nν / , (5.1) N

§ Note that here we neglect the dependence of the normalization constant on t1,t2,...t16, which only weakly eﬀects solution for the most likely state

10 4 where nν are the measured coincidence counts and = ν=1 nν . We can determine the errors in sν using the following formula [26] N P 16 ∂sν ∂sµ δsν δsµ = δnλδnκ, (5.2) ∂nλ ∂nκ λ,κ X=1 where the over-bar denotes the ensemble average of the random uncertainties δsν and δnλ. The measured coincidence counts nλ are statistically independent Poissonian random variables, which implies the following relation:

δnλδnκ = nλδλ,κ, (5.3) where δλ,κ is the Kronecker delta. Taking the derivative of eq.(5.1), we ﬁnd that

∂sµ 1 nµ = δµν 2 Dν , (5.4) ∂nν − N N where

4 1 if 1 ν 4 D = δ = ≤ ≤ (5.5) ν λ,ν  λ=1 0 if 5 ν 16. X  ≤ ≤ Substituting from eq.(5.4) into eq.(5.2) and using eq.(5.3 ), we obtain the result

nµ nν nµ δsν δsµ = δν,µ + (1 Dµ Dν ). (5.6) 2 3 − − N N In most experimental circumstances 1, and so the second term in eq.(5.6) is negligibly small in comparison to the ﬁrst. We shall therefore ignore it,N and ≫ use the approximate expression in the subsequent discussion;

nµ sµ δsν δsµ δν,µ δν,µ. (5.7) ≈ 2 ≡ N N

B. Errors due to Angular Settings Uncertainties

Using the formula (3.7) for the parameters sν we can ﬁnd the dependence of the measured density matrix on errors in the tomographic states. The derivative of sν with respect to some generic wave-plate settings angle θ is

∂sν ∂ ∂ = ψν ρˆ ψν + ψν ρˆ ψν , (5.8) ∂θ ∂θ h | | i h | ∂θ | i where ψν is the ket of the ν-th projection state [see eq.(3.5)]. Substituting from eq.(3.14) we ﬁnd | i 16 ∂sν ∂ ∂ = sµ ψν Mˆµ ψν + ψν Mˆµ ψν . (5.9) ∂θ ∂θ h | | i h | ∂θ | i µ=1 X

For convenience, we shall label the four waveplate angles h ,ν, q ,ν ,h ,ν, q ,ν which specify the ν-th state by { 1 1 2 2 } θν, ,θν, ,θν, ,θν, respectively. Clearly the µ-th state does not depend on any of the ν-th set of angles. Thus we { 1 2 3 4} obtain the following expression for the derivatives of sν with respect to waveplate settings:

16 ∂sν (i) = δν,λ sµfν,µ, (5.10) ∂θλ,i µ=1 X where

(i) ∂ ∂ f = ψν Mˆµ ψν + ψν Mˆµ ψν . (5.11) ν,µ ∂θ h | | i h | ∂θ | i ν,i ν,i

11 (i) The 1024 quantities fν,µ can be determined by taking the derivatives of the functional forms of the tomographic states given by eqs.(3.4) and (3.5), and evaluating those derivatives at the appropriate values of the arguments (see Table 1). The errors in the angles are assumed to be uncorrelated, as would be the case if each wave-plate were adjusted for each of the 16 measurements. In reality, for qubit experiments, only one or two of the four waveplates are adjusted between one measurement and the next. However the assumption of uncorrelated angular errors greatly simpliﬁes the calculation (which is, after all, only an estimate of the errors), and seems to produce reasonable ﬁgures for our error bars ∗∗ Thus with the following assumption 2 δθν,iδθµ,j = δν,µδi,j (∆θ) , (5.12) (where ∆θ is the RMS uncertainty in the setting of the waveplate, with an estimated value of 0.25o for our apparatus) we obtain the following expression for the errors in sν due to angular settings:

4 16 (i) (i) δsν δsµ = δν,µ fν,ǫfν,λsǫsλ (5.13) i=1 ǫ,λ X X=1 Combining eqs.(5.13) and (5.7) we obtain the following formula for the total error in the quantities sν :

δsν δsµ = δν,µΛν (5.14) where

4 16 sν (i) (i) Λν = + f f sǫsλ . (5.15)  ν,ǫ ν,λ  i=1 ǫ,λ N X X=1   (i) These sixteen quantities can be calculated using the parameters sν and the constants fν,ǫ. Note that the same result 2 can be obtained by assuming a priori that the errors in the sν are all uncorrelated, with Λν = δsν ; the more rigorous treatment given here is however necessary to demonstrate this fact. For a typical number of counts, say = it is found that the contribution of errors from the two causes is roughly comparable; for larger numbers ofN counts,∞′′′′ the angular settings will become the dominant source of error. Based on these results, the errors in the values of the various elements of the density matrix estimated by the linear tomographic technique described in Section 3 are as follows:

16 2 ∂ρi,j ∂ρi,j (∆ρi,j ) = δsν δsµ ∂sν ∂sµ ν,µ=1 X 16 2 = Mν(i,j) Λν (5.16) ν=1 X ˆ where Mν(i,j) is the i, j element of the matrix Mν . A convenient way in which to estimate errors for a maximum likelihood tomographic technique (rather than a linear tomographic technique) is to employ the above formulae, with the slight modiﬁcation that the parameter sν should be recalculated from eq.(3.7) using the estimated density matrixρ ˆest. This does not take into account errors inherent in the maximum likelihood technique itself.

C. Errors in Quantities Derived from the Density Matrix

When calculating the propagation of errors, it is actually more convenient to use the errors in the sν parameters, (given by eq.(5.15), rather than the errors in the elements of density matrix itself (which have non-negligible correlations).

∗∗In other experimental circumstances, such as the measurement of the joint state of two spin 1/2 particles, the tomography would be realized by performing unitary operations on the spins prior to measurement. In this case, an assumption analogous to ours will be wholly justiﬁed.

12 1. von Neumann Entropy

The von Neumann entropy is an important measure of the purity of a quantum stateρ ˆ. It is deﬁned by [27]

= Tr ρˆlog (ˆρ) S − { 2 } 4 = pa log pa, (5.17) − 2 a=1 X where pa is an eigenvalue ofρ ˆ, i.e.

ρˆ φa = pa φa , (5.18) | i | i

φa being the a-th eigenstate (a= 1,. . . 4). The error in this quantity is given by | i 16 2 2 ∂ (∆ ) = S Λν . (5.19) S ∂sν ν=1 X Applying the chain rule, we ﬁnd

4 ∂ ∂pa ∂ S = S . (5.20) ∂sν ∂sν ∂pa a=1 X The partial diﬀerential of a eigenvalue can be easily found by perturbation theory. As is well known (e.g. [28]) the change in the eigenvalue λa of a matrix Wˆ due to a perturbation in the matrix δWˆ is

δλa = φa δWˆ φa , (5.21) h | | i where φa is the eigenvector of Wˆ corresponding to the eigenvalue λa. Thus the derivative of λa with respect to some variable| xiis given by

∂λa ∂Wˆ = φa φa . (5.22) ∂x h | ∂x | i 16 ˆ Sinceρ ˆ = ν=1 Mν sν , we ﬁnd that

P ∂pa = φa Mˆ ν φa (5.23) ∂sν h | | i and so, taking the derivative of eq.(5.17), eq.(5.20) becomes

4 ∂ [1 + ln pa] S = φa Mˆ ν φa . (5.24) ∂sν − h | | i ln 2 a=1 X Hence

16 4 2 2 [1 + ln pa] (∆ ) = φa Mˆ ν φa Λν . (5.25) S h | | i ln 2 ν=1 a=1 ! X X For the experimental example given above, =0.106 0.049. S ±

2. Linear Entropy

The “linear entropy” is used to quantify the degree of mixture of a quantum state in an analytically convenient form, although unlike the von Neumann entropy it has no direct information theoretic implications. In a normalized form (deﬁned so that its value lies between zero and one), the linear entropy for a two qubit system is deﬁned by:

13 4 = 1 Tr ρˆ2 P 3 − 4 4 = 1 p2 . (5.26) 3 − a a=1 ! X To calculate the error in this quantity, we need the following partial derivative:

4 ∂ 8 ∂pa P = pa ∂sν −3 ∂sν a=1 X 4 8 = pa φa Mˆ ν φa −3 h | | i a=1 X 8 = Tr ρˆMˆ ν −3 16n o 8 = Tr Mˆ µMˆ ν sµ (5.27) −3 µ=1 X n o Hence the error in the linear entropy is

16 2 2 ∂ (∆ ) = P Λν , P ∂sν ν=1 X 2 16 8 16 = Tr Mˆ Mˆ s Λ . (5.28) 3 µ ν µ ν ν µ=1 ! X X n o For the example given in Sections III and IV, =0.037 0.026. P ±

3. Concurrence, Entanglement of Formation and Tangle

The concurrence, entanglement of formation and tangle are measures of the quantum-coherence properties of a mixed quantum state [29]. For two qubits †† , concurrence is defined as follows: consider the non-Hermitian matrix Rˆ =ρ ˆΣˆˆρTΣˆ where the superscript T denotes transpose and the “spin flip matrix” Σˆ is defined by:

0 00 1 0 01− 0 Σ=ˆ . (5.29)  0 10 0   100 0   −    Note that the deﬁnition of Σˆ depends on the basis chosen; we have assumed here the “computational basis” HH , HV , V H , V V . In what follows, it will be convenient to write Rˆ in the following form: {| i | i | i | i} 1 16 Rˆ = qˆ s s , (5.30) 2 µ,ν µ ν µ,ν=1 X ˆ ˆ ˆ T ˆ ˆ ˆ ˆ T ˆ ˆ whereq ˆµ,ν = MµΣMν Σ+ MνΣMµ Σ. The left and right eigenstates and eigenvalues of the matrix R we shall denote by ξa , ζa and ra, respectively, i.e.: h | | i

ξa Rˆ = ra ξa h | h | Rˆ ζa = ra ζa . (5.31) | i | i

††The analysis in this subsection applies to the two qubit case only. Measures of entanglement for mixed n-qubit systems are a subject of on-going research: see, for example, [30] for a recent survey.

14 We shall assume that these eigenstates are normalized in the usual manner for bi-orthogonal expansions, i.e. ξa ζb = h | i δa,b. Further we shall assume that the eigenvalues are numbered in decreasing order, so that r1 r2 r3 r4. The concurrence is then deﬁned by the formula ≥ ≥ ≥

C = Max 0, √r √r √r √r { 1 − 2 − 3 − 4} 4 3 = Max 0, sgn a √ra , (5.32) 2 − ( a=1 ) X where sgn(x) = 1 if x > 0 and sgn(x) = 1 if x < 0. The tangle is given by T = C2 and the Entanglement of Formation by −

1+ √1 C2 E = h − , (5.33) 2 ! where h(x)= xlog2x (1 x)log2(1 x). Because h(x) is a monotonically increasing function, these three quantities are to some extent− equivalent− − measures− of the entanglement of a mixed state. To calculate the errors in these rather complicated functions, we must employ the perturbation theory for non- Hermitian matrices (see Appendix C for more details). We need to evaluate the following partial derivative,

∂C 4 3 1 ∂r = sgn a a ∂sν 2 − 2 ra ∂sν a=1 √ X 4 3 1 ∂Rˆ = sgn a ξa ζa 2 − 2 ra h |∂sν | i a=1 √ X 4 16 3 1 = sgn a ξa qˆµ,ν sµ ζa , (5.34) 2 − 2 ra h | | i a=1 µ=1 √ X X where the function sgn (x) is the sign of the quantity x: it takes the value 1 if x > 0 and 1 if x < 0. Thus sgn(3/2 a) is equal to +1 if a = 1 and 1 if a =2, 3 or 4. Hence the error in the concurrence− is − − 16 2 2 ∂C (∆C) = Λν ∂sν ν=1 X 16 4 16 2 3 1 = sgn a ξa qˆµ,ν sµ ζa Λν . (5.35) 2 − 2 ra h | | i ν=1 a=1 µ=1 √ ! X X X For our example the concurrence is 0.963 0.018. Once we know the error in the concurrence,± the errors in the tangle and the entanglement of formation can be found straightforwardly:

∆T =2C∆C (5.36) C 1+ √1 C2 ∆E = h′ − ∆C, (5.37) √1 C2 2 ! − where h′(x) is the derivative of h(x). For our example the the tangle is 0.928 0.034 and the entanglement of formation is 0.947 0.025. ± ±

VI. CONCLUSIONS

In conclusion, we have presented a technique for reconstructing density matrices of qubit systems, including a full error analysis. We have extended the latter through to calculation of quantities of interest in quantum information, such as the entropy and concurrence. Without loss of generality, we have used the example of polarization qubits of entangled photons, but we stress that these techniques can be adapted to any physical realization of qubits.

15 ACKNOWLEDGEMENTS

The authors would like to thank Joe Altepeter, Mauro d’Ariano, Zdenek Hradil, Kurt Jacobs, Poul Jessen, Michael Neilsen, Mike Raymer, Sze Tan, and Jaroslav Reh´aˇcekˇ for useful discussions and correspondence. This work was supported in part by the U.S. National Security Agency, and Advanced Research and Development Activity (ARDA), by the Los Alamos National Laboratory LDRD program and by the Australian Research Council.

APPENDIX A: THE Γˆ-MATRICES

One possible set of Γ-matricesˆ are generators of SU(2) SU(2), normalized so that the conditions given in eq.3.8 are fulﬁlled. These matrices are: ⊗ 0100 0 i 0 0 10 0 0 0010 1000 i −0 0 0 0 10 0 0001 Γˆ = 1 , Γˆ = 1 , Γˆ = 1 , Γˆ = 1 , 1 2  0001  2 2  0 0 0 i  3 2  00− 1 0  4 2  1000  −  0010   0 0 i 0   0 0 0 1   0100       −            0001 0 0 0 i 00 1 0 0 0 i 0 0010 0 0 i −0 0 0 0 1 00− 0 i Γˆ = 1 , Γˆ = 1 , Γˆ = 1 , Γˆ = 1 , 5 2  0100  6 2  0 i 0 0  7 2  10 0− 0  8 2  i 00− 0  −  1000   i 0 0 0   0 10 0   0 i 0 0       −            00 0 i 0 00 1 0 0 i 0 100 0 0 0 i −0 0 01− 0 00− 0 i 010 0 Γˆ = 1 , Γˆ = 1 , Γˆ = 1 , Γˆ = 1 , 9 2  0 i −0 0  10 2  0 10 0  11 2  i 0 00  12 2  0 0 1 0  −  i 00 0   100 0   0 i 0 0   00 0 1     −   −   −          01 0 0 0 i 0 0 10 00 1000 10 0 0 i −0 00 0 1 0 0 0100 Γˆ = 1 , Γˆ = 1 , Γˆ = 1 , Γˆ = 1 . 13 2  00 0 1  14 2  00 0 i  15 2  0− 0 1 0  16 2  0010  − −  0 0 1 0   0 0 i 0   00 01   0001   −   −             (A.1)

As noted in the text, this is only one possible choice for these matrices, and the ﬁnal results are independent of the choice.

APPENDIX B: THE Mˆ -MATRICES AND SOME OF THEIR PROPERTIES

The Mˆ matrices, deﬁned by eq.(3.15), are as follows:

16 2 (1 i) (1 + i) 1 0 (1 i)0 1 (1 + i)− 0− − i 0 (1 + i)− 2− i (1 + i) Mˆ = 1 , Mˆ = 1 , 1 2  −(1 i) i 0 0  2 2  − 0 i 0− 0  − − − −  1 0 00   1 (1 + i)0 0     −      00 0 1 0 0 (1 + i) 1 0 0 i (1 + i) 0 0 − i 0 Mˆ = 1 , Mˆ = 1 , 3 2  0 i 0 −(1 i)  4 2  (1 i) i 2 (1 i)  − − − − − − − −  1 (1 i) (1 + i) 2   1 0 (1 + i) 0   − − −   −      0 0 2i (1 + i) 0 0 0 (1 + i) 0 0 (1 i)− 0 0 0 (1 i)− 2i Mˆ = 1 , Mˆ = 1 , 5 2  2i (1 + i)0− 0  6 2  0 (1+ i)0− 0  −  (1 i)0 0 0   (1 i) 2i 0 0   − −   − − −      0 0 0 (1 + i) 0 0 2 (1 + i) 0 0 (1 i)− 2 0 0 (1 i)− 0 Mˆ = 1 , Mˆ = 1 , 7 2  0 (1 + i)0− − 0  8 2  2 (1 + i)0− − 0  − −  (1 i)2 0 0   (1 i)0 0 0   − −   − −      (B.1) 0 0 0 i 0001 0 0 i 0 0010 Mˆ = , Mˆ = , 9  0 i −0 0  10  0100   i 0 0 0   1000   −        0 00 i 0 2 0 (1 + i) 0 0 i 0 2 0 (1 + i)− 0 Mˆ = , Mˆ = 1 , 11  0 i 0 0  12 2  0 (1 i)0− 0  − − −  i 0 00   (1 i)0 0 0   −   − −      0 0 0 (1 + i) 0 0 0 (1 i) 0 0 (1 + i)− 0 0 0 (1 i)− 0− Mˆ = 1 , Mˆ = 1 , 13 2  0 (1 i)0− 2  14 2  0 (1 + i)− 0− 2i  − − − −  (1 i)0 2 0   (1 + i)0 2i 0   − −   −      0 2i 0 (1 i) 00 01 2i −0 (1 i)− 0− 0 0 1 0 Mˆ = 1 , Mˆ = . 15 2  0 (1+ i)0− 0  16  0 1− 0 0  −  (1 + i)0 0 0   10 00   −        The form of these matrices is independent of the chosen set of matrices Γˆν used to convert the density matrix into a column vector. However the Mˆ ν matrices do depend on the set of tomographicn o states ψν . There are some useful properties of these matrices which we will now derive. From eq.(3.15),| i we have

1 ψµ Mˆ ν ψµ = ψµ Γˆλ ψµ B− . (B.2) h | | i h | | i λ,ν λ X

From eq.(3.12) we have ψµ Γˆλ ψµ = Bµ,λ, thus we obtain the result h | | i

ψµ Mˆ ν ψµ = δµ,ν . (B.3) h | | i If we denote the basis set for the four-dimensional Hilbert space by i (i =1, 2, 3, 4) , then eq.(3.14) can be written as follows: {| i }

i ρˆ j = i Mˆ ν j ψν k l ψν k ρˆ l . (B.4) h | | i h | | ih | ih | ih | | i k,l ν X X

17 Since eq.(B.4) is valid for arbitrary statesρ ˆ, we obtain the following relationship:

i Mˆ ν j ψν k l ψν = δikδjl. (B.5) ν h | | ih | ih | i X Contracting eq.(B.5) over the indices (i, j) we obtain:

Tr Mˆ ν ψν ψν = I,ˆ (B.6) ν | ih | X n o where Iˆ is the identity operator for our four dimensional Hilbert space. A second relationship can be obtained by contracting eq.(B.5), viz:

i Mˆ ν j = δij , (B.7) ν h | | i X or, in operator notation,

Mˆ ν = I.ˆ (B.8) ν X

APPENDIX C: PERTURBATION THEORY FOR NON-HERMITIAN MATRICES

Whereas perturbation theory for Hermitian matrices is covered in most quantum mechanics text-books, the case of non-Hermitian matrices is less familiar, and so we will present it here. The problem is, given the eigenspectrum of a matrix Rˆ0 [31], i.e.:

ξa Rˆ = ra ξa (C.1) h | 0 h | Rˆ ζa = ra ζa , (C.2) 0| i | i where

ξa ζb = δa,b (C.3) h | i we wish to ﬁnd expressions for the eigenvalues r′ and eigenstates ξ′ and ζ′ of the perturbed matrix Rˆ′ = Rˆ + δRˆ. a h a| | ai 0 We start with the standard assumption of perturbation theory, i.e. that the perturbed quantities ra′ , ξa′ and ζa′ can be expressed as power series of some parameter λ: h | | i

(0) (1) 2 (2) ra′ = ra + λra + λ ra + . . . (C.4) (0) (1) 2 (2) ζa′ = ζa + λ ζa + λ ζa + . . . (C.5) | i | (0)i | (1)i 2 | (2)i ξ′ = ξ + λ ξ + λ ξ + . . . (C.6) h a| h a | h a | h a |

Writing Rˆ′ = Rˆ0 + λδRˆ, and comparing terms of equal powers of λ in the eigen equations, one obtains the following formulae:

Rˆ ζ(0) = r(0) ζ(0) (C.7) 0| a i a | a i ξ(0) Rˆ = r(0) ξ(0) (C.8) h a | 0 a h a | Rˆ r(0)Iˆ ζ(1) = δRˆ r(1) ζ(0) (C.9) 0 − a | a i − − a | a i ξ(1) Rˆ r(0)Iˆ = ξ(0) δRˆ r(1) . (C.10) h a | 0 − a −h a | − a Equations (C.7) and (C.8) imply that, as might be expected,

(0) ζa = ζa (C.11) | (0)i | i ξa = ξa (C.12) h (0)| h | ra = ra (C.13)

18 Taking the inner product of eq.(C.9) with ξa , and using the bi-orthogonal property eq.(C.3), we obtain h | (1) r = ξa δRˆ ζa . (C.14) a h | | i This implies that

δra r′ ra ≡ a − ξa δRˆ ζa . (C.15) ≈ h | | i Thus, dividing both sides by some diﬀerential increment δx and taking the limit δx 0, we obtain →

∂ra ∂Rˆ = ξa ζa . (C.16) ∂x h | ∂x | i ˆ ˆ Using the completeness property of the eigenstates, b ζb ξb = I and the identity R0 = b rb ζb ξb we obtain the following formula | ih | | ih | P P 1 − 1 Rˆ0 raˆI = ζb ξb . (C.17) − rb ra | ih | b − Xb=a 6 Applying this to eq.(C.9) we obtain

[1] G. C. Stokes, Trans. Cambr. Phil. Soc. 9, 399 (1852). [2] U. Leonhardt, Measuring the quantum state of light (Cambridge University Press, 1997). [3] Z. Hradil, “Quantum-state estimation,” Phys. Rev. A55, R1561 (1997). [4] S. M. Tan, “An inverse problem approach to optical homodyne tomography,” J. Mod. Opt. 44, 2233 (1997). [5] K. Banaszek, G. M. D’Ariano, M. G. A. Paris and M. F. Sacchi, “Maximum-likelihood estimation of the density matrix,” Phys. Rev. A 61, 010304 (1999). [6] Z. Hradil, J. Summhammer, G. Badurek and H. Rauch, “Reconstruction of the spin state,” Phys. Rev. A 62, 014101 (2000). [7] J. Reh´aˇcek,ˇ Z. Hradil and M. Jeˇzek, “Iterative algorithm for reconstruction of entangled states”, quant-ph/0009093 (submitted). [8] J.R. Ashburn, R.A. Cline, P.J.M. Vanderburgt, W.B. Westerveld and J.S. Risley, “Experimentally determined density- matrices for H(n=3) formed in H+-He collisions from 20 to 100 keV,” Phys. Rev. A 41, 2407-2421 (1990). [9] D. T. Smithey, M. Beck, M. G. Raymer and A. Faridani, “Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum,” Phys. Rev. Lett. 70, 1244 (1993). [10] G. Breitenbach, S. Schiller, J. Mlynek, “Measurement of the quantum states of squeezed light,” Nature 387, 471-475 (1997).

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20 ^ ^ ^ ^ 1 qubit µ0 µ1 µ2 µ3

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 2 qubits µ0 µ1 µ2 µ3 µ0 µ1 µ2 µ3 µ0 µ1 µ2 µ3 µ0 µ1 µ2 µ3

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 3 qubits ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3 µ0µ1µ2µ3

• ...... • FIG. 1. Tree diagram representing number and type of measurements necessary for tomography. For a single qubit, the measurements {µˆ0, µˆ1, µˆ2, µˆ3} suﬃce to reconstruct the state, e.g., measurements of the horizontal, vertical, diagonal and right-circular polarization components, (H,V,D,R). For two qubits, 16 double-coincidence measurements are necessary ({µˆ0µˆ0, µˆ0µˆ1 . . . µˆ3µˆ3}), increasing to 64 three-coincidence measurements for three qubits ({µˆ0µˆ0µˆ0, µˆ0µˆ0µˆ1 . . . µˆ3µˆ3µˆ3}), and so on, as shown.

photon detector polarizer

“Black Box” Coincidence source of photon pairs in QWP HWP arbitrary quantum states Detector output

polarizer photon detector FIG. 2. Schematic illustration of the experimental arrangement. “QWP” stands for quarter waveplate, “HWP” for half waveplate; the angles of both pairs of wave plates can be set independently giving the experimenter four degrees of freedom with which to set the projection state. In the experiment, the polarizers were realized using polarizing prisms, arranged to transmit vertically polarized light.

21 0.6 0.6

0.4 0.4

0.2 HH 0.2 HH 0 HV 0 HV VH VH HH HH VH VV VH VV HH HH VH VH

0.6 0.6

0.4 0.4

0.2 HH 0.2 HH 0 HV 0 HV VH VH HH HH VH VV VH VV HH HH VH VH FIG. 3. Graphical representation of the density matrix of a state as estimated by linear tomography (left) and by maximum likelihood tomography (right) from the experimental data given in the text. The upper plot is the real part ofρ ˆ, the lower plot the imaginary part.

22 ν Mode 1 Mode 2 h1 q1 h2 q2 1 H H 45o 0 45o 0 2 |Hi |Vi 45o 0 0 0 3 |Vi |Vi 0 0 0 0 4 |Vi |Hi 0 0 45o 0 5 |Ri |Hi 22.5o 0 45o 0 6 |Ri |Vi 22.5o 0 0 0 7 |Di |Vi 22.5o 45o 0 0 8 |Di |Hi 22.5o 45o 45o 0 9 |Di |Ri 22.5o 45o 22.5o 0 10 |Di |Di 22.5o 45o 22.5o 45o 11 |Ri |Di 22.5o 0 22.5o 45o 12 |Hi |Di 45o 0 22.5o 45o 13 |Vi |Di 0 022.5o 45o 14 |Vi |Li 0 022.5o 90o 15 |Hi |Li 45o 0 22.5o 90o 16 |Ri |Li 22.5o 0 22.5o 90o Table| i 1 | i

TABLE 1: The tomographic analysis states used in our experiments. The number of coincidence counts measured in projections measurements provide a set of 16 data that allow the density matrix of the state of the two modes to be estimated. We have used the notation D ( H + V ) /√2, L ( H + i V ) /√2 and R ( H i V ) /√2. Note that, when the measurement are taken| i≡ in| thei order| i given| byi≡ the| table,i | onlyi one waveplate| i≡ | anglei− had| i to be changed between each measurement.